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Rotational Symmetry of Quadrilaterals by Beverly Heigre

Rotational Symmetry of Quadrilaterals by Beverly Heigre
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2015-06-15 16:05
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This GeoGebra example dynamically demonstrates the rotational symmetry of quadrilaterals, showing how different quadrilaterals (such as squares, rectangles, rhombi) coincide with their original shape after rotation by certain angles. The core mathematical concepts include rotational symmetry, center of rotation, angle of rotation, geometric transformations, and properties of polygons. The construction uses GeoGebra commands such as Segment, Polygon, Circle, Rotate, and Mirror to create the quadrilateral and its rotated copies, while OrthogonalLine, Intersect, and Point are used to determine the rotation center and symmetry axes. Additional objects like Line and Tangent Function enhance the analysis. This resource is suitable for middle and high school geometry classes, helping students grasp the idea of rotational symmetry and explore how different quadrilaterals exhibit varying degrees of symmetry, thereby fostering spatial reasoning and logical thinking.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. OrthogonalLine(A, a) → b
  3. OrthogonalLine(B, a) → c
  4. Point(b) → C
  5. OrthogonalLine(C, b) → d
  6. Intersect(c, d) → D
  7. Polygon(C, D, B, A) → poly1, c_1, d_1, b_1, a_1
  8. Segment(C, B) → e
  9. Segment(D, A) → f
  10. Intersect(e, f) → E
  11. Polygon(F, G, 4) → poly2, f_1, g, h, i, H, I
  12. Segment(I, G) → j
  13. Segment(H, F) → k
  14. Intersect(j, k) → J
  15. Segment(K, L) → l
  16. Circle(K, L) → p
  17. Point(p) → M
  18. Segment(K, M) → m
  19. Line(M, l) → n
  20. Line(L, m) → q
  21. Intersect(n, q) → N
  22. Segment(M, L) → r
  23. Segment(N, K) → s
  24. Intersect(r, s) → O
  25. Polygon(M, N, L, K) → poly3, m_1, n_1, l_1, k_1
  26. Rotate(C, (-α), E) → C'
  27. Rotate(D, (-α), E) → D'
  28. Rotate(B, (-α), E) → B'
  29. Rotate(A, (-α), E) → A'
  30. Polygon(C', D', B', A') → poly1', c', d', b', a'
  31. Rotate(M, (-α), O) → M'
  32. Rotate(N, (-α), O) → N'
  33. Rotate(L, (-α), O) → L'
  34. Rotate(K, (-α), O) → K'
  35. Polygon(M', N', L', K') → poly3', m', n', l', k'
  36. Segment(P, Q) → t
  37. Segment(P, R) → e_1
  38. Line(Q, e_1) → g_1
  39. Line(R, t) → h_1
  40. Intersect(g_1, h_1) → S
  41. Polygon(P, R, S, Q) → poly4, p_1, r_1, s_1, q_1
  42. Segment(R, Q) → i_1
  43. Segment(S, P) → j_1
  44. Intersect(i_1, j_1) → T
  45. Rotate(P, (-α), T) → P'
  46. Rotate(R, (-α), T) → R'
  47. Rotate(S, (-α), T) → S'
  48. Rotate(Q, (-α), T) → Q'
  49. Polygon(P', R', S', Q') → poly4', p', r', s', q'
  50. Segment(U, V) → t_1
  51. Segment(U, W) → a_2
  52. Line(W, t_1) → b_2
  53. Point(b_2) → Z
  54. Segment(Z, V) → c_2
  55. Polygon(U, V, Z, W) → poly5, u, v, z, w
  56. Segment(W, V) → d_2
  57. Segment(Z, U) → e_2
  58. Intersect(d_2, e_2) → A_1
  59. Rotate(U, (-α), A_1) → U'
  60. Rotate(V, (-α), A_1) → V'
  61. Rotate(Z, (-α), A_1) → Z'
  62. Rotate(W, (-α), A_1) → W'
  63. Polygon(U', V', Z', W') → poly5', u', v', z', w'
  64. Line(B_1, C_1) → f_2
  65. Mirror(D_1, f_2) → E_1
  66. Polygon(D_1, B_1, E_1, C_1) → poly6, d_3, b_3, e_3, c_3
  67. Segment(D_1, E_1) → g_2
  68. Intersect(f_2, g_2) → F_1
  69. Rotate(D_1, (-α), F_1) → G_1
  70. Rotate(B_1, (-α), F_1) → H_1
  71. Rotate(E_1, (-α), F_1) → I_1
  72. Rotate(C_1, (-α), F_1) → J_1
  73. Polygon(G_1, H_1, I_1, J_1) → poly6', g_3, h_2, i_2, j_2
  74. Rotate(poly2, (-α), J) → poly2'

AI Drawing Prompt Reference

Draw a quadrilateral demonstrating rotational symmetry. Include the center of rotation, rotation angle, and the rotated quadrilateral. The construction order is: first draw the quadrilateral, determine the rotation center, then rotate the vertices, and finally display the rotated figure to show how different quadrilaterals coincide with their original shape after rotation.
The prompt above was compiled from the original drawing commands and is for reference only.